Question

What points, if any, are the functions discontinuous? $$ g(u)=\frac{u^{2}+|u-1|}{\sqrt[3]{u+1}} $$

Short answer

The function is continuous for all real numbers; there are no points of discontinuity.

Step-by-step solution

Identify Types of Discontinuities

There are three main types of discontinuities: removable jump, and infinite. We will check which of these appear in the given function, if any.

Check Domain of g(u)

The function is defined for values of \(u\) where the denominator \(\sqrt[3]{u+1}\) does not cause undefined values. The cube root is defined for all real numbers, so no restrictions arise from \(u + 1\).

Analyze the Numerator for Discontinuity

The numerator, \(u^2 + |u-1|\), is the sum of a quadratic function and an absolute value function. Both of these are continuous for all real numbers, so they do not introduce discontinuities.

Conclude Discontinuity Points

Since neither the numerator nor the cube root of the denominator introduces any discontinuities and they are both defined for all real numbers, the function \(g(u)\) is continuous everywhere within its domain.

Key concepts

Types of Discontinuities

To understand where a function might be discontinuous, it's important to grasp the different types of discontinuities that may occur. There are three main types:

• Removable Discontinuity: This occurs when a function has a "hole" at a certain point. The limit exists at that point, but the actual function value doesn't match it.
• Jump Discontinuity: This kind of discontinuity happens when a sudden "jump" exists in the function's value from one side of a point to the other, resulting in no single limit.
• Infinite Discontinuity: Present when the function approaches infinity at a specific point, creating a vertical asymptote.

Identifying these is key to analyzing any function for continuity or points of discontinuity. It's always helpful to inspect the behavior of both the numerator and the denominator of the fractions in your function to see which discontinuities might affect it.

Domain of a Function

The domain of a function is the set of all possible inputs (or 'x' values) for which the function is defined. Understanding the domain is crucial as it helps us know where the function can "live" and behave as expected. For functions involving denominators, like fractions, a key step is to ensure the denominator doesn't become zero, as this would make the function undefined.
In the case of the function \( g(u) = \frac{u^2 + |u-1|}{\sqrt[3]{u+1}} \), the denominator, a cube root function \( \sqrt[3]{u+1} \), is defined for all real numbers. This means that the cube root, unlike a square root, doesn't become undefined for any real number. Therefore, our function \( g(u) \) doesn't have any restrictions on its domain due to the denominator.
Knowing the domain helps avoid points that are undefined or might cause discontinuities, simplifying the continuity analysis.

Numerator Analysis

Analyzing the numerator of a function can provide insights into potential discontinuities. For our function \( g(u) = \frac{u^2 + |u-1|}{\sqrt[3]{u+1}} \), the numerator is made up of two parts: a quadratic function \( u^2 \) and an absolute value function \( |u-1| \).
Quadratic functions are smooth and continuous across all real numbers. They don't introduce any discontinuities by themselves.
The term \( |u-1| \) represents an absolute value function, which is also continuous everywhere in its domain. Absolute value functions can create sharp turns, but they do not disrupt the continuity in a mathematical sense. This means the numerator in our function remains continuous over all real numbers, contributing no discontinuities to \( g(u) \).
By confirming the continuity of the numerator, we ensure that any discontinuities must instead arise from the denominator of the function.

Continuous Functions

A function is said to be continuous if there are no breaks, jumps, or "holes" in its graph. It smoothly flows from one point to another, without interruptions. Determining whether a function is continuous involves checking both the numerator and the denominator for possible discontinuities.
For our function \( g(u) \):- Neither the numerator \( u^2 + |u-1| \) nor the denominator \( \sqrt[3]{u+1} \) introduces discontinuities.
The cube root in the denominator is defined for all real numbers, adding smooth continuity. The numerator, comprising a continuous quadratic term and an absolute value term, remains unbroken throughout its domain.
Since every part of the function \( g(u) \) is continuously defined for all real numbers, the function itself is continuous everywhere. Identifying continuity in functions is crucial as it helps us understand the behavior of functions across their entire domain, ensuring that they behave in a predictable and reliable manner.